Iterate MapThread with matrices

Question

Consider a function f and a list p={a,b,c}. I want to get a list of

f[a,a]
f[a,b]
f[a,c]
f[b,a]
f[b,b]
f[b,c]
f[c,a]
f[c,b]
f[c,c]

In real use, p can have higher dimensions, and f may take more arguments, e.g. I may need to generate a list of f[a,a,a,a] through f[z,z,z,z]. Is there a cleaner way to do this other than making lists that approximately repeat the elements of p then use MapThread? e.g.

p={a,b}
p1 = {a, a, b, b}
p2 = {a, b, a, b}
MapThread[f, {p1, p2}]

Here, a, b can be matrices

Answer 1

You can use Tuples:

Tuples[f[{a, b, c}, {a, b, c}]]

{f[a, a], f[a, b], f[a, c], f[b, a], f[b, b], f[b, c], f[c, a], f[c, b], f[c, c]}

f @@@ Tuples[{a, b, c}, 2]

same result

Tuples[f[{a, b}, {r, s, t}, {x, y}]]

{f[a, r, x], f[a, r, y], f[a, s, x], f[a, s, y], f[a, t, x], f[a, t, y], f[b, r, x], f[b, r, y], f[b, s, x], f[b, s, y], f[b, t, x], f[b, t, y]}

f @@@ Tuples[{{a, b}, {r, s, t}, {x, y}}]

same result

p = {a, b};
p1 = {a, a, b, b};
Tuples[f[p, p1]]

{f[a, a], f[a, a], f[a, b], f[a, b], f[b, a], f[b, a], f[b, b], f[b, b]}

Answer 2

You could also use Tuples as follows:

Tuples[f[a,b,c], 2]

{f[a, a], f[a, b], f[a, c], f[b, a], f[b, b], f[b, c], f[c, a], f[c, b], f[c, c]}

If f evaluates, and you want to do this for matrices, you could do:

p = RandomInteger[1, {3, 2, 2}];
Block[{KroneckerProduct}, Tuples[KroneckerProduct @@ p, 2]] //Dimensions

{9, 4, 4}